Reduction of lateral pressure propagation due to dissipation into ambient mudrocks during geological carbon dioxide storage

Authors


Abstract

[1] Carbon dioxide (CO2) storage in deep geological formations can lead to significant reductions in anthropogenic CO2 emissions if large amounts of CO2 can be stored. Estimates of the storage capacity are therefore essential to the evaluation of individual storage sites as well as the feasibility of the technology. One important limitation on the storage capacity is the radius of review, the lateral extent of the pressure perturbation, of the storage project. We show that pressure dissipation into ambient mudrocks retards lateral pressure propagation significantly and therefore increases the storage capacity. For a three-layer model of a reservoir surrounded by thick mudrocks, the far-field pressure is approximated well by a single-phase model. Through dimensional analysis and numerical simulations, we show that the lateral extent of the pressure front follows a power law that depends on a single dissipation parameter math formula, where math formula and math formula are the ratios of mudrock to reservoir permeability and specific storage, and math formula is the aspect ratio of the confined pressure plume. Both the coefficient and the exponent of the power law are sigmoid decreasing functions of math formula. The math formula values of typical storage sites are in the region where the power-law changes rapidly. The combination of large uncertainty in mudrock properties and the sigmoid shape leads to wide and strongly skewed probability distributions for the predicted radius of review and storage capacity. Therefore, if the lateral extent of the pressure front limits the storage capacity, the determination of the mudrock properties is an important component of site characterization.

1. Introduction

[2] Geological storage of math formula has been proposed in saline aquifers, deep sea sediments, depleted oil and gas reservoirs, and in combination with enhanced oil recovery or enhanced coalbed methane recovery [Intergovernmental Panel on Climate Change (IPCC), 2005]. To make a significant contribution to the mitigation of climate change, gigatonnes of math formula must be stored in the subsurface every year, and therefore, large regional saline aquifers are the primary target for geological math formula storage [Metz et al., 2006].

[3] To decrease the storage volume, math formula is injected at depths greater than 800 m, where math formula is supercritical for most geotherms [Holloway and Savage, 1993]. Supercritical math formula is less dense than the brine under most continental and shallow marine storage conditions [Bachu, 2008], and therefore, a low-porosity and low-permeability seal that provides a capillary entry barrier for the math formula is necessary to prevent direct upward migration of buoyant supercritical math formula. Evidence that a typical seal has blocked upward migration of the injected math formula out of the storage formation is provided by repeated seismic surveys at the Sleipner math formula-injection site [Chadwick et al., 2005].

[4] The viability of a math formula storage project hinges on its storage capacity, the amount of math formula that can be injected into the storage formation. Many studies have been carried out to assess the capacity of sedimentary basins to store math formula [Myer et al., 2005; Bradshaw et al., 2007; Kopp et al., 2009]. The estimate of a basin-scale storage capacity is based on the effective pore volume of the target formation [Bachu and Adams, 2003], but the storage capacity is also affected by heterogeneity, residual saturation of each phase, pressure buildup, math formula migration, and the injection scenario [Bachu, 2008; Hesse et al., 2008]. This study is focused on the limitation due to pressure buildup in the reservoir. For the purpose of this study, the pressure buildup, or the overpressure, is defined as the increase of the reservoir pressure above the preinjection pressure. Thibeau and Mucha [2011] argue that either local or global pressure buildup is the primary control on the storage capacity and that a simple volumetric capacity assessment overestimates the storage capacity. Szulczewski et al. [2012] argue that pressure buildup is the limiting factor for short injection times while math formula migration would be limiting for longer injection times.

[5] Hydraulically closed formations are not suitable for large injection volumes [Ehlig-Economides and Economides, 2010], which require open, permeable, thick, and laterally continuous regional aquifers to allow the displacement of in situ brine by the injected math formula [Van der Meer, 1992; Holloway et al., 1996; Doughty and Pruess, 2004]. Even in open formations, the math formula storage capacity may be limited by local pressure buildup near wells or by basin-scale (10–100 km) pressure buildup [Thibeau and Mucha, 2011]. Geomechanical studies show that low injectivity leads to local pressure buildup near the injector that can cause the geomechanical failure of the surrounding seals [Rutqvist and Tsang, 2002; Rutqvist et al., 2007; Birkholzer et al., 2009; Zhou et al., 2009; Rutqvist et al., 2010; Morris et al., 2011; Vilarrasa et al., 2011]. The limit placed on the injection operation due to local pressure buildup could be mitigated by the addition of injectors that reduce the maximum pressure by distributing the injected math formula over a larger area. In contrast, Thibeau and Mucha [2011] point out that regional scale pressure buildup resulting from the displacement of formation fluids cannot be mitigated by the addition of injectors within the same field. Figure 1 shows the comparison of the pressure perturbation induced by either a single or multiple well(s), which inject the same total amount of fluid, into an identical reservoir with properties given in Table 1. In the scenario with multiple injectors, the maximum of pressure buildup at the center is greatly reduced relative to the single injector scenario, but the radial extent of the pressure buildup is nearly identical. This illustrates that it is easier to manage the local pressure buildup with optimal design of the injection operation than to control its regional extent, which mainly reflects the total fluid mass injected (similar math formula). Recently, it has been recognized that the radial extent of the pressure perturbation may therefore be an important constraint on the overall storage capacity [Birkholzer and Zhou, 2009; Birkholzer et al., 2011a] and requires the definition of a radius of review math formula for a geological math formula storage site. To establish a radius of review math formula, a pressure cutoff has to be defined. The area inside the radius of review, where the pressure increases above this cutoff, is considered to be affected by the injection operation. Physical processes that have been considered in the definition of the pressure cutoff are brine displacement, capillary entry pressure, and geomechanical failure.

Table 1. Summary of Model Properties
 PropertiesFigure 1Figures 5 and 6Figure 7
  1. a

    If a parameter was varied systemically, it is indicated by “Var.”, and given in the legend or axes of the respective figure.

Mudrock math formula(m math formula)  math formulaVar.
math formula(Pa math formula) 1.22 math formula10 math formulaVar.
math formula 0.350.35
math formula(m) 3,0003,000
math formula(m) 10,00010,000
Sandstone math formula(m math formula)5.43 math formula10 math formula5.43 math formula10 math formula5.43 math formula10 math formula
math formula(Pa math formula)6.15 math formula10 math formula6.15 math formula10 math formula6.15 math formula10 math formula
math formula0.250.250.25
math formula(m)12510, 25, 50, 75, 100
math formula(m)10,00010,00010,000
Fluid math formula(Pa math formula)4.0 math formula10 math formula4.0 math formula10 math formula4.0 math formula10 math formula
math formula (Pa s)1 math formula10 math formula1 math formula10 math formula1 math formula10 math formula
Figure 1.

The effect of the number of injection wells on the pressure field is shown. In both cases, CO2 was injected at the same total injection rate of 10 Mt/yr for 30 years. The sandstone reservoir has math formula m2 and math formula Pa−1 given in Table 1. The white line represents the radial extent of the pressure front set by the cutoff value of math formula.

[7] Nicot et al. [2006] argue that the pressure cutoff for the radius (area) of review of geological math formula storage should be determined based on the possibility of contaminating nearby potable aquifers. This is similar to an earlier definition developed by Thornhill et al. [1982], which defines the radius of review as the area in which injection-induced overpressure may cause migration of the injected or preexisting formation fluids into potable groundwater resources. Nicot et al. [2008] and Birkholzer et al. [2011b] argue that such leakage is most likely through a permeable conduit, i.e., a borehole or a preexisting fracture. They define the pressure cutoff as the minimum value above which sustained migration of formation fluids into the potable aquifer is induced. Based on a static mass balance, they determine a pressure cutoff in the range from 0.1 to 0.6 MPa for a typical reservoir at a depth of 1.5 km. Their approach assumes that the sealing unit contains a preexisting permeable conduit, which lowers the pressure cutoff compared to an intact seal. The other constraint can be the math formula capillary entry pressure into the sealing unit, which is a gradient of 1.9 MPa/km [Springer and Lindgren, 2006]. One geomechanical aspect is that the hydraulic fracturing of the sealing unit can limit the maximum injection-induced overpressure ranging between 3.5 and 8 MPa/km [Thibeau and Mucha, 2011]. The geomechanical and capillary entry pressure constraints can be addressed by a suitable design of the injection scenario, and here we focus on the pressure cutoff imposed by potential brine displacement. Therefore, the lower pressure cutoffs for brine displacement are relevant for the definition of the radius of review. Here we choose a pressure cutoff of 1 MPa, somewhat larger than the values cited above, which consider the extreme case of an open leakage path. Section 2.4 shows that the definition of the radius of review is not sensitive to the absolute value of the pressure cutoff, as long as it is a small fraction of the characteristic injection pressure defined in section 2.3. The absolute value chosen for the pressure cutoff therefore does not affect the results of this study.

[8] It is common for math formula storage studies to focus only on the target formation and to exclude ambient mudrocks, defined as a fine- to very fine-grained siliciclastic sediments or sedimentary rocks [Grainger, 1984] bounding the reservoir because low-permeability mudrocks prevent vertical math formula migration due to high capillary entry pressure. However, most regional aquifers are not closed and also the overlying and underlying mudrocks are not perfectly impermeable [Neuzil, 1994; Dewhurst et al., 1999]. Pressure buildup caused by injection may partially dissipate into and through these units even if the capillary entry pressure does not permit math formula flow across the boundary as shown in Figure 2. The vertical pressure communication between layers mostly depends on the vertical permeability and specific storage of the seals [Domenico and Schwartz, 1998; Hovorka et al., 2001; Hart et al., 2006; Zhou et al., 2008; Birkholzer et al., 2009; Chadwick et al., 2009; Szulczewski et al., 2012].

Figure 2.

Basin-scale and pore-scale illustration of diffusion of injection-induced overpressure (red line) as well as convection of injected fluids (blue line). Even though injected fluids may not migrate into the ambient mudrock due to low permeability, pressure can be absorbed into the mudrock due to relatively high specific storage.

[9] Neuzil [1994] shows that both laboratory and regional studies give the effective permeability of the sealing unit between math formula and math formula, and small-scale fractures may increase the permeability. However, in many realistic field sites, the seal will not be a uniform mudrock. This study was partly motivated by the Cranfield site, where the sealing unit is sedimentologically heterogeneous with complex and fine interlayering of laterally discontinuous mudrocks and coarser sediments [Lu et al., 2011]. Because spatial distribution of these fine details are neither known nor is it possible to resolve them numerically, they have to be represented by increased effective mudrock permeability.

[10] The second important physical property affecting the pore fluid pressure is the specific storage of the system due to the rock and fluid compressibilities. The relatively few data for the specific storage at greater depth are reviewed below. The hysteretic specific storage has to be kept in mind when the unloading during geological math formula storage is compared with the loading of the aquifer during water withdrawal. The impact of hysteresis on the specific storage decreases with increasing depth of burial and we are not aware of data that constrain it at depth. Therefore, the hysteretic effect is neglected in this analysis, but the likely effect of hysteresis on the results is discussed in section 5.2.

[11] Rieke and Chilingarian [1974] obtain compressibility equations as a function of pressure that shows that clay-rich mudrocks have compressibilities up to 2 orders of magnitude higher than fluids at the depth of up to 3 km for math formula storage as well as 1–3 orders of magnitude higher than the reservoir rock itself, which ranges from math formula to math formula at depths greater than 1 km [Ge and Garven, 1992]. This suggests that the ambient permeable and compressible mudrock may contribute significantly to the dissipation of the injection pressure as shown in Figure 2.

[12] Simulations of geological math formula storage that include the ambient mudrocks often assume they have small compressibilities and negligible permeabilities to avoid a significant impact on the pressure evolution in the reservoir [Rutqvist and Tsang, 2002; Audigane et al., 2007; Van der Meer and Van Wees, 2006; Rutqvist et al., 2007; Vilarrasa et al., 2011]. In contrast, many hydrological studies consider the pressure response in layers adjacent to the pumped aquifer, and they show that groundwater withdrawal from a confined aquifer induces a significant pressure perturbation in the pumped aquifer as well as adjacent mudrocks [Wolff, 1970; Hsieh, 1996; Leake and Hsieh, 1997; Burbey, 2001; Muggeridge et al., 2005]. Table 2 summarizes the petrophysical properties of ambient mudrocks used in previous modeling studies of geological math formula storage and some hydrological studies for comparison. Despite the differences in aquifer depth and loading conditions, the range of mudrock specific storage used in the literature of geological math formula storage is broadly comparable with the hydrological studies, but the range of mudrock permeabilities assumed in studies of geological math formula storage is significantly lower.

Table 2. Physical Properties of the Sandstone Reservoir and Ambient Mudrock for the Previous Studies math formula
Previous studies math formula math formula(Pa) math formula(Pa) math formula math formula math formula(Pa) math formula(Pa math formula) math formula math formula(Pa math formula) math formula(m math formula)
  1. a

    Properties in the first five literature are used in the hydrological models, and those in bottom 14 literature are used in the math formula storage models.

  2. b

    Drained vertical bulk modulus defined as math formula.

  3. c

    Mudrock compressibility of a Frio-type reservoir is from Riney et al. [1985].

  4. d

    Mudrock compressibility of a Cranfield-type reservoir is from Chierici et al. [1978].

Ge and Garven [1992] math formula2.0 math formula10 math formula 0.32.7 math formula10 math formula3.7 math formula10 math formula0.31.7 math formula10 math formula3.2 math formula10 math formula
math formula6.0 math formula10 math formula 0.26.7 math formula10 math formula1.5 math formula10 math formula0.21.0 math formula10 math formula3.2 math formula10 math formula
Hsieh [1996] math formula 3.0 math formula10 math formula0.259.0 math formula10 math formula1.1 math formula10 math formula0.41.1 math formula10 math formula1.0 math formula10 math formula
math formula 3.0 math formula10 math formula0.259.0 math formula10 math formula1.1 math formula10 math formula0.31.2 math formula10 math formula1.0 math formula10 math formula
Leake and Hsieh [1997] math formula8.0 math formula10 math formula 0.259.6 math formula10 math formula1.0 math formula10 math formula0.41.1 math formula10 math formula1.2 math formula10 math formula
math formula8.0 math formula10 math formula 0.259.6 math formula10 math formula1.0 math formula10 math formula0.41.2 math formula10 math formula3.0 math formula10 math formula
Burbey [2001] math formula 1.2 math formula10 math formula0.253.6 math formula10 math formula2.8 math formula10 math formula0.12.8 math formula10 math formula5.9 math formula10 math formula
math formula 1.2 math formula10 math formula0.253.6 math formula10 math formula2.8 math formula10 math formula0.252.9 math formula10 math formula5.9 math formula10 math formula
Muggeridge et al. [2005] math formula    1.0 math formula10 math formula0.051.0 math formula10 math formula1.0 math formula10 math formula
math formula    1.0 math formula10 math formula0.25.0 math formula10 math formula1.0 math formula10 math formula
Hovorka et al. [2001] math formula    1.5 math formula10−9c0.11.5 math formula10−91.0 math formula10 math formula
math formula    1.5 math formula10 math formula0.32.6 math formula10 math formula4.0 math formula10 math formula
Audigane et al. [2007] math formula    4.6 math formula10 math formula0.18.7 math formula10 math formula1.0 math formula10 math formula
math formula    1.9 math formula10 math formula0.426.2 math formula10 math formula3.0 math formula10 math formula
Rutqvist et al. [2007] math formula5.0 math formula10 math formula 0.256.0 math formula10 math formula1.7 math formula10 math formula0.011.7 math formula10 math formula1.0 math formula10 math formula
math formula5.0 math formula10 math formula 0.256.0 math formula10 math formula1.7 math formula10 math formula0.12.1 math formula10 math formula1.0 math formula10 math formula
Doughty et al. [2008] math formula    1.5 math formula10−9c0.11.5 math formula10 math formula1.0 math formula10 math formula
math formula    4.0 math formula10 math formula0.311.4 math formula10 math formula2.5 math formula10 math formula
Zhou et al. [2008] math formula    5.4 math formula10 math formula0.121.0 math formula10 math formula1.0 math formula10 math formula
math formula    5.4 math formula10 math formula0.121.0 math formula10 math formula1.0 math formula10 math formula
Birkholzer et al. [2009] math formula    4.5 math formula10 math formula0.056.5 math formula10 math formula1.0 math formula10 math formula
math formula    9.0 math formula10 math formula0.25.3 math formula10 math formula1.0 math formula10 math formula
Vidal-Gilbert et al. [2009] math formula1.4 math formula10 math formula 0.291.8 math formula10 math formula5.7 math formula10 math formula0.057.7 math formula10 math formula1.0 math formula10 math formula
math formula2.4 math formula10 math formula 0.293.1 math formula10 math formula3.2 math formula10 math formula0.159.2 math formula10 math formula1.0 math formula10 math formula
Zhou et al. [2009] math formula    1.1 math formula10 math formula0.151.7 math formula10 math formula1.0 math formula10 math formula
math formula    4.5 math formula10 math formula0.124.2 math formula10 math formula1.0 math formula10 math formula
Ferronato et al. [2010] math formula    2.1 math formula10 math formula0.271.3 math formula10 math formula1.0 math formula10 math formula
math formula    2.1 math formula10 math formula0.271.3 math formula10 math formula2.0 math formula10 math formula
Rutqvist et al. [2010] math formula2.0 math formula10 math formula 0.152.1 math formula10 math formula4.7 math formula10 math formula0.015.1 math formula10 math formula1.0 math formula10 math formula
math formula6.0 math formula10 math formula 0.26.7 math formula10 math formula1.5 math formula10 math formula0.172.2 math formula10 math formula1.3 math formula10 math formula
Morris et al. [2011] math formula5.0 math formula10 math formula 0.36.7 math formula10 math formula1.5 math formula10 math formula0.11.9 math formula10 math formula1.0 math formula10 math formula
math formula1.0 math formula10 math formula 0.21.1 math formula10 math formula9.0 math formula10 math formula0.171.6 math formula10 math formula1.3 math formula10 math formula
Preisig and Prevost [2011] math formula2.0 math formula10 math formula 0.152.1 math formula10 math formula4.7 math formula10 math formula0.015.1 math formula10 math formula1.0 math formula10 math formula
math formula6.0 math formula10 math formula 0.26.7 math formula10 math formula1.5 math formula10 math formula0.172.2 math formula10 math formula1.3 math formula10 math formula
Vilarrasa et al. [2011] math formula5.0 math formula10 math formula 0.36.7 math formula10 math formula1.5 math formula10 math formula0.011.5 math formula10 math formula1.0 math formula10 math formula
math formula1.0 math formula10 math formula 0.31.3 math formula10 math formula7.4 math formula10 math formula0.11.1 math formula10 math formula1.0 math formula10 math formula
Choi et al. [2011] math formula    1.2 math formula10−8d0.351.2 math formula10 math formula2.9 math formula10 math formula
math formula    6.2 math formula10 math formula0.257.2 math formula10 math formula2.7 math formula10 math formula

[17] In section 2, we present a dimensional analysis that shows that the ratios of permeabilities and specific storage of the sandstone reservoir and the ambient mudrocks ( math formula and math formula) are the governing parameters determining the pressure evolution in a layered formation. Figure 3 shows the values of math formula and math formula used in the previous studies of hydraulic pumping and math formula storage. We can see that the ratios math formula and math formula for math formula storage sites vary by 6 and 3 orders of magnitude, respectively. This suggests that current math formula injection sites shown in Figure 3 span the range of almost no dissipation of overpressure to a considerable dissipation into the mudrocks and may undergo a large range of possible reservoir responses to injection-induced overpressure. In this manuscript, we present a comprehensive analysis of the effect of pressure dissipation into ambient mudrocks on the lateral pressure propagation, the radius of review, and the storage capacity. In particular, we highlight the role of the mudrock-specific storage, which can be significantly different from that of the reservoir ( math formula).

Figure 3.

The data of math formula and math formula from the previous studies of pressure diffusion in a layered system for hydraulic pumping and geological CO2 storage. The formation properties for each study are given in Table 2. The diagonal lines represent the same ratio of hydraulic diffusivities math formula.

[18] For the purpose of this study, we model the rock volume in which math formula injection is taking place as an infinite horizontal sandstone layer overlain by infinitely thick mudrocks that exhibit various combinations of permeability ( math formula to math formula) and specific storage ( math formula to math formula) given in Table 1. We perform a scaling analysis and a simulation study of this simplified layered model to provide a pressure history throughout the storage formation as well as the surrounding mudrocks. Hence, we calculate the radius of review, the storage capacity, and the volume fraction of the brine displaced into the ambient mudrock as a function of the petrophysical parameters of both layers, the thickness of the reservoir, and the injection rate and duration.

2. Model Problem

[19] The injection of math formula into a brine-filled reservoir leads to a two-phase flow near the injector. Nicot et al. [2009] argue that at late times and large distances the pressure disturbance created by the single- and two-phase injection will be similar. Figure 4 shows that the pressure profiles from single- and two-phase flow in radial geometry match well outside the region directly invaded by the math formula. This confirms that single-phase pressure is a good approximation for the pressure evolution beyond the region under the effect of two-phase flow. However, the pressure at the well is significantly lower due to the increase in total mobility with increasing math formula saturation. Here we focus on regional-scale pressure necessary to define the radius of review and therefore neglect two-phase flow near the injection well.

Figure 4.

Comparison of radial single- and two-phase flow models. (a) Pressure profiles at three time steps. The intersecting points with the dimensionless pressure cutoff ( math formula) determines the location of the pressure front at each time. (b) The position of the pressure front with time. For the region beyond the effect of two-phase flow, both pressure profiles are identical.

2.1. Governing Equations

[20] The equation for pore pressure dissipation in single-phase flow in a porous medium is obtained from the equation of mass conservation and Darcy's law. Assuming constant external stresses and uniaxial deformation in the vertical direction, we study the pressure evolution using a single-phase pressure diffusion model. In cylindrical coordinates, math formula, centered on the injection well, the flow of a slightly compressible fluid in a heterogeneous and isotropic compressible porous medium is described by the diffusion equation

display math(1)

and the initial condition

display math(2)

where math formula is the overpressure defined as the amount of pore pressure math formula exceeding the hydrostatic pressure math formula is the total compressibility, math formula is the permeability, math formula is the porosity, and math formula (Pa s) is the fluid viscosity. Under the assumptions of incompressible solid grains and pores as well as uniaxial strain, the total compressibility is defined as

display math(3)

where math formula (Pa−1) and math formula (Pa−1) are the fluid and rock bulk compressibilities, respectively [Van der Kamp and Gale, 1983; Green and Wang, 1990]. For the range of data given in Table 2, the rock compressibilities for both sandstone and mudrock are several orders of magnitude larger than the fluid compressibility, which we assume to be constant and equal to math formula Pa−1 [Freeze and Cherry, 1979]. The left side of equation (1) suggests the introduction of the uniaxial specific storage S (Pa math formula) given by Wang [2000].

display math(4)

[21] An interesting result of this approximation of the total compressibility and the small fluid compressibility is that the specific storage S is essentially independent of porosity. The model properties are summarized in Table 1, and the reference cases for the application of the results are defined using physical properties from the math formula injection sites: In Salah (Krechba, Algeria), Frio (TX, USA), and Cranfield (MS, USA). To facilitate comparison, we are considering a generic injection scenario that does not reproduce the particular conditions at either one of these sites; therefore, we refer to them below as In Salah-type, Frio-type, and Cranfield-type sites or reservoirs.

2.2. Layered Mudrock-Sandstone Geometry

[22] To study the effect of pressure dissipation into ambient formations from a single well, we consider a three-layer geometry comprising a laterally extensive horizontal sandstone reservoir overlain and underlain by thick mudrocks, assumed to be thick enough to contain the entire vertical extent of the pressure plume, and the domain is therefore radially and vertically infinite. We use a cylindrical coordinate system with the origin at the location of the injector and centered on the sandstone reservoir. In this geometry, overpressure is symmetric across a horizontal plane at the center of the reservoir so that only the positive z axis has to be considered (Figure 5). Assuming fluid injection at math formula with constant rate Q (m math formula/s), the boundary conditions are given by

display math(5)

where math formula (m) is the half thickness of the sandstone reservoir in z direction and math formula (yrs) is the injection period. The spatial variation of the physical properties in this layered geometry is given by

display math(6)

where the subscripts s and m denote sandstone and mudrock, respectively.

Figure 5.

(a) A typical simulation grid used in this study. At the boundary between the two layers and near the well, finer grids resolve the strong pressure gradient. (b) A pressure profile for the case of math formula. The white line represents the pressure contour line of 1 Pa of math formula corresponding to math formula. (c) Pressure contour lines corresponding to different math formula by changing permeability of mudrock math formula, while keeping math formula and math formula fixed at nonzero values; math formula and math formula. An impermeable mudrock ( math formula) confines injection-induced overpressure perfectly within the sandstone reservoir, and hence, math formula approaches 1.5 shown as a red line.

2.3. Scaling Analysis

[23] To develop a dimensionless form of equation (1), we define dimensionless variables as

display math(7)

with the characteristic scales given by

display math(8)

where math formula is a scale for the pressure buildup due to injection and math formula is a scale for the lateral pressure propagation in the reservoir. Then, we can rewrite equation (1) in dimensionless form as follows

display math(9)

[24] The corresponding initial and boundary conditions are

display math(10)

where math formula is the dimensionless well radius. The dimensionless parameter fields as a function of space are given by

display math(11)

[25] Accordingly, the dimensionless problem (9)–(11) has only three independent governing parameters defined as

display math(12)

where math formula (m math formula/s) is hydraulic diffusivity of the sandstone layer defined as

display math(13)

[26] Using (13), we can define another parameter, the ratio of hydraulic diffusivities math formula, which can be expressed by math formula and math formula

display math(14)

[27] The effect of math formula on the pressure dissipation in a layered system will be discussed in section 5.3.

2.4. Limiting Radial Solution

[28] If math formula and either math formula or math formula approaches zero, the problem reduces to the solution of Theis [1935], which is self-similar in the Boltzmann variable math formula and given by

display math(15)

[29] At late time and large distance from the well, math formula, Van Everdingen and Hurst [1949] show that the pressure front is approximated by

display math(16)

[30] Previous studies of pressure diffusion have suggested that the self-similarity between math formula and math formula can explain the propagation of the pressure front in the radial semi-infinite domain [Van Poolen, 1964; Talwani and Acree, 1984; Shapiro et al., 1997; Daungkaew et al., 2000]. From the solution (16), it follows that the radial distance from the well to the pressure front, i.e., radius of review, given by

display math(17)

[31] The coefficient math formula is defined by

display math(18)

where math formula (Pa) is the cutoff amount for the pressure increase. In the limit of math formula, the coefficient math formula approaches 1.5 at which math formula is not considered to be an independent parameter. In radial coordinates, the maximum dimensionless radius of review, math formula, will be math formula. Our numerical solutions to the radial problem show good agreement with the analytical solution (17). Later we show that dissipation into ambient mudrocks affects this first-order scaling law for the lateral propagation of the pressure pulse. Although the propagation follows a power law in all cases considered, the coefficient math formula and, more importantly, the exponent decreases with increasing dissipation.

3. Numerical Results

[32] We perform finite-element flow simulations of the model defined by the governing equation (9), the boundary and initial conditions (10), for a large range of the dimensionless parameters (11) to study the effect of pressure dissipation on lateral pressure propagation. The analytic solution for the confined case (17) gives the maximum distance of pressure diffusion, and the computational domain was chosen much larger, so that the outer boundary conditions had no effect on the solution. The finite-element analysis is conducted with COMSOL Multiphysics [2011] using bilinear quadrilateral elements for spatial discretization [Hughes, 2000] and a variable step method for time integration [Dreij et al., 2011]. We use numerical grids that are highly refined near the boundary of the reservoir as shown in Figure 5a to resolve the strong pressure gradient typical for this problem, as in the pressure field shown in Figure 5b.

3.1. Effect of Permeability Ratio math formula

[33] Before considering the full problem governed by all three parameters math formula, and math formula, we illustrate the effect of pressure confinement or dissipation on the power law by varying math formula while keeping math formula and math formula fixed at values typical for Cranfield-type reservoir, i.e., math formula and math formula. To compare multiple pressure plumes, we just show the pressure contour equal to the cutoff value math formula corresponding to math formula. Figure 5c shows how increasing mudrock permeability allows pressure dissipation and reduces the speed of lateral pressure propagation.

[34] Figure 6a shows that the propagation of the pressure front with dissipation into the mudrock remains linear on a log-log plot and is still given by a power law. To quantify these dissipative losses, we assume a general power law for the lateral pressure propagation of the form

display math(19)

where math formula and math formula are functions of math formula, and math formula. In the limit of impermeable surrounding layers, the exponent math formula approaches 0.5 and the coefficient math formula approaches math formula. The exponent math formula corresponds to the slopes in Figure 6a and decreases monotonically with increasing math formula as more overpressure dissipates into the ambient mudrocks as shown in Figure 6b.

Figure 6.

(a) The dimensionless location of the pressure front versus time. The lateral pressure propagation in the numerical simulations follows a power law even with the attenuation of overpressure into the mudrock. The linear regression of the numerical data on the log-log plot generally gives a good fit with a math formula of 0.99. (b) The power-law exponent math formula as a function of math formula, and data from three injection sites are indicated on the plot. The red line indicates the 0.5 exponent of the Theis solution for the confined pressure propagation. (c) The power-law coefficient math formula as a function of math formula. The coefficient depends on the pressure cutoff math formula.

Figure 7.

The variation of the power-law exponent math formula and coefficient math formula in the three-dimensional parameter space spanned by math formula, and math formula. The surface are fits to 480 numerical simulations distributed throughout the parameter space.

[35] The dependence of math formula and math formula on the dimensionless pressure cutoff math formula is also shown in Figures 6b and 6c. Analogous to the radial analytical solution the power-law exponent math formula in the layered problem is independent of the cutoff. The coefficient math formula is reduced with increasing math formula similar to the decay of math formula given by equation (18). Figure 6c shows that the reduction of math formula is strongest in the limit of math formula, where math formula approaches math formula. As noted earlier, if math formula is at its asymptotic value of 1.5 and similarly math formula is constant.

3.2. Dissipation Parameter for a Layered System

[36] The scaling analysis presented in section 2.3 suggests that the power law (19) should be a function of the individual ratios of mudrock to reservoir permeability and specific storage ( math formula and math formula) and the aspect ratio of the maximum radial diffusive distance to the reservoir thickness ( math formula). To analyze the effect of each parameter on the pressure evolution, we performed 480 simulations for a parameter study varying math formula from math formula to math formula from math formula to math formula, and math formula from math formula to math formula. These ranges for math formula and math formula correspond to the rock properties reported in the literature (Figure 3) and the range of math formula corresponds to 30 year injection into a reservoir of thickness math formula varying from 10 to 100 m. We define the location of the pressure front using a pressure cutoff math formula so that math formula, and the dimensionless location of the pressure front with time follows the power law in all simulations.

[37] The numerical results are summarized in Figure 7, which shows the variation of the coefficient math formula and the exponent math formula as a function of the governing parameters math formula, and math formula. Both math formula and math formula decrease with increasing math formula, and math formula, which implies that higher math formula, and math formula result in more retardation of the pressure front in a sandstone reservoir. In other words, more overpressure will be attenuated into mudrocks due to either the high permeability of the mudrock (large math formula) or the high specific storage of the mudrock (large math formula) or due to a large aspect ratio of the pressure plume, and hence a large surface area across which overpressure can leak into the mudrock (large math formula).

[38] Figure 7 shows that the contour surfaces of both math formula and math formula form a set of parallel planes in the three-dimensional parameter space of math formula, and math formula. The main result of this study is that the planar contours in Figure 7 show that the power law for pressure propagation depends on a single dissipation parameter defined by the normal to these planes. The normal is given by

display math(20)

and an arbitrary point math formula in the parameter space. We define a dissipation parameter M that will collapse the three variables into a single one:

display math(21)

where ( math formula) is the arbitrary origin set to the minimum values of the governing parameters (−8, 0, 2) in this study. Figures 8a and 8b show that all data for math formula and math formula collapse to a single line if plotted as a function of M, and that M varies more than 2 orders of magnitude. Figure 8 shows that math formula and math formula are monotonically decreasing functions of M. Two plateaus, where the power law is a weak function of M, are separated by a sharp transition between M of 1.5 and 4. To obtain expressions for math formula and math formula that can be evaluated we fit numerical data in Figure 8 with generalized logistic functions

display math(22)
display math(23)
Figure 8.

(a and b) Both math formula and math formula are decreasing functions of M and the fits with the logistic functions given by equations (22) and (23). The maximum of math formula varies depending on math formula. The three injection sites are indicated as circles. (c) The volume fraction of the displaced fluid into ambient mudrock using equation (30). (d and e) The distribution of the probability density function (PDF) of M used for the uncertainty quantification in section 4.3.

[39] Together with the expression for M in terms of the three governing parameters math formula, and math formula, equations (22) and (23) completely determine the power law for the lateral propagation of the dimensionless pressure.

4. Application to math formula Storage

4.1. Reduction of the Radius of Review

[40] The radius of review can be calculated as the radius of the outer extent of the pressure front. Sample calculations of the radius of review are performed for math formula injection with constant rate of 10 Mt of math formula per year, which is equivalent to math formula of math formula per year, assuming math formula density math formula is 600 kg/m math formula at average pressure P = 16 MPa and temperature T = 60°C [Bachu, 2003], into a sandstone reservoir surrounded by mudrocks for 30 years. For the cases considered below, a characteristic pressure math formula is 200 MPa for In Salah-type, 8 MPa for Frio-type, and 5 MPa for Cranfield-type reservoirs in which math formula varies up to 0.2 so that math formula is in the range 1.2 to 1.5 as shown in Figure 8b. The In Salah-type reservoir has a large math formula because of relatively low permeability of the reservoir math formula. In this case, the maximum pressure near the well would have to be reduced by injection through multiple wells similar to Figure 1b or horizontal wells not to trigger hydraulic fracture. Injection through n suitably placed vertical wells would reduce the characteristic pressure to math formula. Furthermore, in a two-phase flow system, i.e., math formula injection into brine-saturated reservoir, the actual maximum overpressure will be lower than math formula due to the higher mobility of math formula phase as pointed out in Figure 4a.

[41] To illustrate the effect of pressure dissipation, we use the geometry and petrophysical properties of In Salah-type, Frio-type, and Cranfield-type reservoirs, summarized in Table 3. To facilitate comparison and to highlight the effect of pressure dissipation, we assume that the same total amount of math formula (0.3 Gt equivalent to math formula) is injected at each site.

Table 3. Estimates of the Radius of Review and the Storage Capacity
 PropertiesIn Salah-TypeFrio-TypeCranfield-Type
  1. a

    The standard physical properties of each reservoir, i.e., math formula and math formula, are from Morris et al. [2011], Hovorka et al. [2001], and Choi et al. [2011] given in Table 2.

  2. b

    Data from Michael et al. [2010].

  3. c

    The radius of review from the radial solution (17) with math formula = 30 yrs.

  4. d

    The percentage error of math formula from the preferred value math formula.

  5. e

    The preferred value calculated using the standard properties of each reservoir.

  6. f

    The amount of injected CO2 at 10 Mt/yr for 30 years, which excludes the capacity of the ambient mudrocks.

  7. g

    The percentage error of math formula from the preferred value math formula.

Dimensionless parameterslog math formula−5.1−6.6−3.0
log math formula0.10.81.2
log math formula2.83.42.8
M1.852.123.19
math formula1.51.31.2
Power-law parameters math formula0.420.380.28
math formula1.190.840.17
Physical properties math formula math formula(m)151232
math formula (m math formula/s)8.2 math formula10 math formula1.5 math formula10 math formula3.8 math formula10 math formula
Radius of review estimate math formulac(km)13.2 (+25.2%) math formula51.3 (+58.4%)23.1 (+632.5%)
math formulae(km)10.5132.363.15
Storage capacity estimate math formulaf(Gt)0.3 (−45.5%)g0.3 (−74.3%)0.3 (−99.9%)
math formula(yrs)55.1136.833153.4
math formulae(Gt)0.551.37331.53

[49] Using the data of each reservoir and the information of the injection operation as shown in Table 3, we can evaluate the dimensionless parameter M as defined in equation (21). The values of M vary from 1.85 for an In Salah-type reservoir to 3.19 for a Cranfield-type reservoir. Although the relative mudrock permeability math formula is lowest at the Frio-type site, it has a higher M value and is therefore more susceptible to pressure dissipation than In Salah-type site, because the relative specific storage of the mudrock math formula is higher and the aspect ratio of the confined pressure plume math formula is larger. Both In Salah- and Cranfield-type reservoirs have similar geometry, i.e., math formula is similar, but the M value of the Cranfield-type reservoir is much higher because of larger permeability and specific storage of the surrounding mudrocks. Larger permeability at the Cranfield-type reservoir represents an estimate of the effective value for the heterogeneous mudrocks.

[50] Given M for these sites, the sigmoid curve fit equations (22) and (23) give the coefficient math formula and the exponent math formula, and using the characteristic scales (8), the radius of review is given by

display math(24)

and is reported in Table 3. The Theis solution (17) gives the upper limit of the radius of review, math formula, which regards ambient mudrocks as perfectly closed boundaries. Even this upper bound varies from 13 km at an In Salah-type site to 51 km at a Frio-type site, mostly due to much higher hydraulic diffusivity math formula at the Frio-type site.

[51] If pressure dissipation into the ambient mudrock is included, all three injection sites see a significant reduction in the radius of review. The In Salah-type site with the lowest M value is least affected, but its radius of review after 30 years of injection is increased by 25%, if an impermeable and incompressible mudrock is assumed. As the M value increases, the use of math formula overestimates the radius of review up to 58% for a Frio-type site. The Cranfield-type reservoir surrounded by highly permeable and compressible mudrocks results in a large reduction of the radius of review, and whether the enormous reduction in the pressure plume size at this last site is realistic is discussed in section 5.5. However, even in the other two cases, the reduction in the radius of review is large enough to significantly increase the storage capacity.

4.2. Estimating Storage Capacity

[52] In the section 4.1, we estimated the radius of review given a certain amount of injection. For the site characterization, the more relevant question may be how much math formula can be injected, i.e., what is the storage capacity, given a maximum possible radius of review at a given site, math formula.

[53] The retardation of the pressure front due to dissipation of overpressure into the ambient mudrocks needs to be considered in the estimate of the storage capacity. Solving the following nonlinear equation, we can estimate the maximum injection period math formula at which the pressure front approaches the given math formula from (19) as

display math(25)

[54] Given data for three sites, the amount of injected math formula (Gt) can be estimated by

display math(26)

and is reported in Table 3. The results confirm that permeable and compressible mudrocks provide a larger math formula storage capacity of the target sandstone reservoir. The In Salah-type reservoir with the lowest M value is the least affected by the ambient mudrocks, but the storage capacity is reduced by 46%, if an impermeable and incompressible mudrock is assumed. For a reservoir with a larger M value, the assumption of incompressible mudrocks underestimates the storage capacity up to 74% for a Frio-type reservoir. Similar to the estimates for the radius of review, the use of Cranfield-type properties leads to unrealistically large increases in the storage capacity that cannot be taken at face value and will be discussed in section 5.5.

4.3. Uncertainty Due to Mudrock Properties

[55] Figure 8 shows that the M values for all three sites are in the transition region (1.5 math formula M math formula 4). As a result, uncertainty in the physical properties leads to significant uncertainty in the radius of review and the storage capacity.

[56] The range of physical properties for mudrocks compiled in Table 2 and illustrated in Figure 3 is very large. This is partly due to the large variability of the rock types that are collected under this category in simulation studies. This reflects the difficulty of measuring representative properties, and in several cases, appropriate data are simply not available and the values are merely estimates. The combination of this variability and uncertainty in the mudrock properties with the sensitive dependence of the power law on M introduces large uncertainty to any estimate of the radius of review. If the storage capacity is limited by the lateral pressure propagation, the uncertainty in the mudrock properties will introduce large uncertainty into estimates of the storage capacity. The discussion of the uncertainty will focus on the In Salah-type and Frio-type sites, because as shown in sections 4.1 and 4.2, the physical properties assumed for a Cranfield-type site give unrealistic results.

[57] To illustrate this uncertainty, we generate 4000 values of permeability and specific storage of mudrock ( math formula and math formula) from a lognormal distribution around the preferred values with a standard deviation of either 0.25 or 1. The resulting distributions of M values (M distribution) from the Monte Carlo method are shown in Figures 8d and 8e remain roughly lognormal, although some skew toward lower M becomes apparent as the variance increases. The effect that uncertainty in M has on the prediction of the radius of review depends on the location of the M distribution relative to the transition zone of the sigmoidal curves for math formula and math formula.

[58] The top row of Figure 9 shows the distributions of the predicted radius of review ( math formula distribution), given a fixed injection volume and injection rate. As expected, the distributions become wider as the variance in the mudrock properties increases. The math formula distributions for In Salah-type sites are both narrower and more skewed than for Frio-type sites. For both sites, the maximum of the math formula distribution is centered on the mean for low variance, but becomes shifted at higher variance. These results can be understood in terms of the shape of the M curve. For the same variance in mudrock properties, the uncertainty in the math formula prediction is larger for the Frio-type site, because the M distribution is located in the center of the transition zone, where the power law for the lateral pressure propagation is a strong function of M. The sigmoid shape of the M curve also explains the skew of the math formula distribution for the In Salah site, where the low tail of the M-distribution samples the left plateau. As the variance increases, the number of realizations sampling the plateau increases until the maximum of the math formula distribution is shifted to larger values. For Frio-type sites, the M distribution begins to sample both plateaus as the variance increases, leading to a bimodal math formula distribution with a minimum near the mean value.

Figure 9.

The effect of uncertainty in mudrock properties ( math formula and math formula) on the estimate of the radius of review math formula and the storage capacity math formula. The mudrock properties are assumed to have a lognormal distribution around the preferred values given in Table 3. Results for a standard deviation of 0.25 are shown as red bars and results for a standard deviation of 1 are shown as blue bars. The orange lines represent the radial diffusive distance excluding the ambient rock properties from the one-dimensional solution (17). The green lines represent math formula and math formula using the preferred values of data from each site. The gray band represents the location of the radial CO2 saturation front. The dashed lines represent the percentage change of math formula and math formula relative to the preferred value.

[59] The bottom row of Figure 9 shows the distributions of the predicted storage capacity ( math formula distribution), given a fixed radius of review and a fixed injection rate. The shape of the math formula distribution depends on the solutions of equation (25) and is more difficult to interpret. Again the uncertainty in the math formula estimate increases with increasing variance in mudrock properties that makes uncertainty for the Frio-type site higher. For both sites, the distributions are centered on the mean and strongly skewed toward higher values.

[60] This simple analysis illustrates the complexity in estimating the uncertainty of the radius of review or the storage capacity. In particular, estimates based on the mean physical properties of the mudrock may be good guide, if the storage capacity has to be evaluated for a given radius of review, because the maximum of the math formula distribution is close to the estimate based on the mean. In contrast, estimates based on the mean physical properties of the mudrock may not be a good guide, if the radius of review has to be evaluated for a desired storage capacity and if the uncertainty in mudrock properties is large, because the maximum of the math formula distribution may be unrelated to the mean estimate.

4.4. Volume of the Displaced Brine Due to Injection

[61] The injection of fluid causes an increase in the storage formation pressure, which will induce the displacement of the preexisting fluid in the formation. For the two-phase flow system of math formula injection into a brine-saturated reservoir surrounded by permeable and compressible mudrocks, injected math formula will be trapped by capillarity, but the brine itself can leak into the ambient mudrock. In our single-phase flow model, we can calculate the amount of the fluid in each layer using the following equations

display math(27)
display math(28)

where math formula (m math formula) is the volume of the fluid, and the subscript i is s for the sandstone reservoir and m for the ambient mudrock. The volume fraction of the total injected fluid that has been displaced into the mudrock math formula is given by

display math(29)

[62] Figure 8c shows that math formula data from the numerical simulations also collapse into a single line as a function of M. To obtain expressions for math formula, we generate the generalized logistic function

display math(30)

[63] As shown in Figure 8c, math formula increases with M because more fluids will be displaced into more permeable (larger math formula) and more compressible (larger math formula) mudrocks surrounding a thinner reservoir (larger math formula). For an In Salah-type site, approximately half of the displaced formation brine will be stored in the ambient rock and even more for a Frio-type site. This highlights that large fluid volumes can be absorbed by mudrocks due to their high specific storage during injection. Again, the Cranfield-type rock properties give unrealistic predictions. This suggests that the applicability of the simple hydrological storage model needs to be evaluated carefully.

5. Discussion

5.1. Scaling Laws

[64] We have demonstrated that dissipation of pressure into ambient mudrocks can significantly reduce the exponent of the power law governing the lateral spreading of the pressure plume. In contrast, the nonlinearities of two-phase flow or phase change near the well do not affect the power law governing the radial pressure propagation [O'Sullivan, 1981] nor does the mechanical deformation of the reservoir due to full poroelastic coupling [Helm, 1994; Hsieh and Cooley, 1995]. In this sense, pressure dissipation into ambient formations has a more profound effect on the large-scale pressure evolution in the reservoir than these other processes that have received most of the attention in the literature on geological math formula storage. The strong effect of dissipative terms on the power law for diffusive propagation is well known and has been discussed extensively in the literature on self-similar solutions. Barenblatt [1996] has shown that the power-law exponent is a continuous function of the dissipative loss, in this case the parameter M, which correlates with the fluid loss from the aquifer.

5.2. Effect of Hysteretic Specific Storage

[65] The increase of pore pressure induced by math formula injection results in a reduction of the effective stress, so that the target reservoir and the ambient mudrocks will be under unloading conditions. The response of sedimentary rocks to loading and unloading can be different, and this is usually described by the hysteretic specific storage given by Hueckel and Nova [1979], Neuzil and Pollock [1983], Neuzil [1993], and Hoffmann et al. [2003]:

display math(31)

where math formula and math formula are the effective and virgin specific storage, and math formula and σ math formula are the effective and preconsolidation stresses. For fine-grained unconsolidated sediments, i.e., mudrock, math formula is up to 2 orders of magnitude smaller than math formula. Most studies of math formula storage have neglected this hysteresis effect because geological math formula storage targets deep, strongly compacted, and diagenetically altered formations, where hysteresis effects become smaller than in shallow groundwater systems [Settari, 2002]. However, some hysteresis is likely to occur and given the lack of detailed geomechanical characterizations of this adds additional uncertainty to the description of the mudrock properties. Qualitatively, the specific storage under elastic unloading is reduced by up to 2 orders of magnitude. This should reduce M values and shift them to the left plateau of the M-curve (1.58 for In Salah-type and 1.52 for Frio-type) at which both math formula and math formula are less sensitive to the uncertainty in M, but the dissipation into the ambient mudrocks will still account for 20% of the total injected fluids.

5.3. Hydraulic Diffusivity Ratio of Layered Systems

[66] Much of the literature on pressure diffusion [Terzaghi, 1943; Bredehoeft and Hanshaw, 1968; Talwani and Acree, 1984; Song et al., 2004; Jaeger et al., 2007] discusses one-dimensional transient-flow problems in a homogeneous domain where math formula, and S can be combined into hydraulic diffusivity math formula. The emphasis on hydraulic diffusivity in the literature may suggest that the ratio of hydraulic diffusivities math formula would govern the lateral pressure propagation in this layered model. Typical values of math formula are in the range of math formula to math formula, as shown in Figure 3.

[67] In heterogeneous media, such as the layered system considered here, equation (1) cannot be rearranged to introduce math formula as a dimensionless parameter. Our results show that the power law for the lateral pressure propagation depends on math formula instead of math formula, and hence lines of constant math formula are perpendicular to lines of constant M as shown in Figure 10a. This means that any math formula and math formula can be obtained from M for a given math formula, i.e., the lateral pressure propagation is independent of the value of math formula. We illustrate this surprising conclusion by comparing two sets of simulations:

Figure 10.

(a) A plane of constant math formula with lines of constant math formula (color) and lines of constant math formula (solid black). We show two sets of simulations to confirm whether math formula can define the lateral pressure propagation or not. The first set has three simulations with a fixed value of math formula but a range of M values. Parameters are shown as circles on the solid line in Figure 10a and the resulting pressure contours in Figure 10b. The second set has three simulations with a fixed value of M = 1.1, but a range of math formula values. Parameters are shown as circles on the dashed line in Figure 10a and the resulting pressure contours in Figure 10c.

[68] (1) Three simulations with a fixed value of math formula, but a range of M values. Parameters are shown as circles on the solid line in Figure 10a, and the resulting pressure contours in Figure 10b.

[69] (2) Three simulations with a fixed value of M = 1.1, but a range of math formula values. Parameters are shown as circles on the dashed line in Figure 10a, and the resulting pressure contours in Figure 10c.

[70] The pressure contours in Figures 10b and 10c show that simulations with constant math formula have variable lateral pressure propagation but constant vertical pressure propagation, while simulations with constant M have constant lateral pressure propagation but variable vertical pressure propagation. The ratio of hydraulic diffusivities therefore controls the vertical propagation of pressure in the layered system while the dissipation parameter M determines the lateral pressure propagation.

[71] The observation that the lateral pressure propagation is independent of math formula may be counterintuitive, if the amount of pressure that has diffused out of the reservoir is equated with the mass of brine that has leaked out of the reservoir. Equation (28) shows that the mass of brine displaced from the reservoir is proportional to the integral of the pressure times the specific storage of the ambient mudrock. Figure 10b shows that math formula controls the vertical pressure profile but the mass of brine leaked from the reservoir also depends on the magnitude of the specific storage. Simultaneously changing math formula and math formula by the same factor does not affect the vertical pressure profile because math formula is constant but it allows arbitrary changes in the mass of brine leaked from the reservoir. Section 4.4 also shows that the fraction of fluid mass leaked from the reservoir is only a function of M and therefore independent of math formula. This clearly illustrates that it is the mass of brine leaked into the mudrock, not the distance of vertical pressure propagation, which determines the speed of the lateral pressure propagation in the reservoir.

5.4. Assumption of Single-Phase Flow

[72] The permeable and compressible ambient mudrock may cause a significant retardation of the pressure front, which may lead to interference between the retarded pressure front and the saturation front that has been neglected in this analysis. We can estimate the radial propagation of the math formula plume, i.e., the saturation front, with time using a simple solution in Buckley-Leverett form [Woods and Comer, 1962]

display math(32)

where math formula is the radius of math formula-rich zone around the injector, math formula is volume fraction factor for math formula is gas saturation for math formula, and math formula is the fractional flow function that measures the gas fraction of the total flow given by

display math(33)

where math formula is the mobility of the gaseous math formula and the formation fluid ( math formula and math formula), math formula is the relative permeability that accounts for the reduced permeability of each phase due to the presence of another phase, and math formula (Pa s) is the viscosity of each phase ( math formula Pa s). The derivative of math formula at the saturation of the front math formula is determined by the tangent construction [Welge et al., 1962]. Bennion and Bachu [2005] give relative permeability data for three sandstone and three carbonate formations that are candidate math formula storage sites in onshore North American sedimentary basins. From this data set, the range of the derivative of math formula at the tangent point is math formula.

[73] The corresponding range in the location of the saturation front after 30 year injection is shown as the gray band in Figures 9a and 9b. For both In Salah- and Frio-type reservoirs, the retarded pressure front corresponding to the mean mudrock properties propagates much further than the saturation front math formula, and the single-phase approach gives a reasonable estimate of the radius of review even with the attenuation of overpressure into the ambient mudrock. If uncertainty in the mudrock properties is large math formula, the low end of the math formula distribution is comparable to the location of the saturation front, and the pressure front interacts with the saturation front. In these cases, the location of the pressure front in the two-phase model is likely to be larger then the estimate based on single-phase flow.

5.5. Limits of the Transient Flow Equation

[74] The transient flow equation (1) is a special case of the more general theory of poroelasticity in the limit of uniaxial strain and constant vertical stress. In hydrologic applications, the difference between them is usually small. Linear poroelasticity itself, however, is a linearized theory that is only applicable for small deformations. In section 4, the Cranfield-type physical properties led to very strong retardation of the pressure pulse and corresponding very large storage capacity. High specific storage of the ambient mudrock allows large volumes of fluid stored in the mudrock, in this case, more than 90% of the injected fluid volume. In reality, however, such large changes in fluid storage are likely to lead to significant deformation that is not accurately described by constant specific storage in a linearized theory.

[75] Accurate predictions of the radius of review and the storage capacity in aquifers surrounded by very compressible mudrocks may therefore require nonlinear geomechanical models that account for changes in specific storage with increasing deformation. Currently, most estimates of the pressure buildup are based on linear theories, and the results need to be evaluated carefully if M is high.

6. Summary and Conclusion

[76] This study shows that ambient mudrocks can have a strong impact on the pressure evolution within a geological math formula storage reservoir. The lateral propagation of the overpressure due to math formula injection is significantly reduced by dissipation into ambient mudrocks.

[77] In a layered system, the pressure evolution is governed by three dimensionless parameters defined by ratios of permeability and specific storage between the reservoir and the ambient mudrock ( math formula and math formula) as well as the aspect ratio of the confined pressure plume ( math formula). Numerical simulations show that the spreading of the pressure front with time in the sandstone reservoir follows a power law even with attenuation of overpressure into the ambient mudrock. The coefficient math formula and the exponent math formula in the power law are governed by a single dissipation parameter math formula. Both the coefficient and the exponent of the power law are sigmoid decreasing function of M. The ratio of hydraulic diffusivities math formula does not describe the lateral pressure propagation, but it is suitable to describe the vertical pressure propagation in the layered system.

[78] A compilation of physical mudrock properties used in recent studies of geological math formula storage shows large variability in the three governing parameters and hence in the dissipation parameter M. This suggests that the importance of pressure dissipation into the mudrock can vary widely between different storage sites. The power law described here enables the quantification of the uncertainty in predictions of the radius of review and the storage capacity using the Monte Carlo method. The M values of typical geological storage sites are in the region where the power law is changing rapidly. In combination with large uncertainty in mudrock properties, this leads to wide and strongly skewed probability distributions for the predicted radius of review and storage capacity.

[79] Our results show that estimates based on the mean physical properties of the mudrock may be a good guide if the storage capacity has to be evaluated for a given radius of review. In contrast, estimates based on the mean physical properties of the mudrock may not be a good guide if the radius of review has to be evaluated for a desired storage capacity. In this case, the mean values are likely to underestimate the radius of review when the uncertainty in mudrock properties is large.

[80] The characterization of the physical properties of the mudrock is an important component of the evaluation of a geological math formula storage site. The effect of pressure dissipation into the ambient mudrocks should be considered in simulation studies of geological math formula storage that aim to determine the pressure evolution.

Acknowledgments

[81] The authors are grateful to the sponsors of the Gulf Coast Carbon Center, Bureau of Economic Geology, The University of Texas at Austin, which partly supported this work and to its Director Susan D. Hovorka. Additional support comes from EPA STAR grant R834384 and the SECARB project, managed by the Southern States Energy Board and funded by the U.S. Department of Energy, NETL, as part of the Regional Carbon Sequestration Partnerships program under contract DE-FC26-05NT42590. The authors gratefully acknowledge anonymous referees of this paper for their constructive comments as well.

Ancillary